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<div class="titlepage"><div><div><h3 class="title">
<a name="boost_multiprecision.tut.gen_int"></a><a class="link" href="gen_int.html" title="Generic Integer Operations">Generic Integer Operations</a>
</h3></div></div></div>
<p>
        All of the <a class="link" href="../ref/number.html#boost_multiprecision.ref.number.integer_functions">non-member
        integer operations</a> are overloaded for the <a href="https://en.cppreference.com/w/cpp/language/types" target="_top">fundamental
        (built-in)</a> integer types in <code class="computeroutput"><span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">multiprecision</span><span class="special">/</span><span class="identifier">integer</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></code>.
        Where these operations require a temporary increase in precision (such as
        for <code class="computeroutput"><span class="identifier">powm</span></code>), then if no <a href="https://en.cppreference.com/w/cpp/language/types" target="_top">fundamental (built-in)</a>
        type is available, a <a class="link" href="ints/cpp_int.html" title="cpp_int">cpp_int</a>
        of appropriate precision will be used.
      </p>
<p>
        Some of these functions are trivial, others use compiler intrinsics (where
        available) to ensure optimal evaluation.
      </p>
<p>
        The overloaded functions are:
      </p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Integer</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">I2</span><span class="special">&gt;</span>
<span class="identifier">Integer</span><span class="special">&amp;</span> <span class="identifier">multiply</span><span class="special">(</span><span class="identifier">Integer</span><span class="special">&amp;</span> <span class="identifier">result</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">I2</span><span class="special">&amp;</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">I2</span><span class="special">&amp;</span> <span class="identifier">b</span><span class="special">);</span>
</pre>
<p>
        Multiplies two <code class="computeroutput"><span class="identifier">I2</span></code> values,
        to produce a wider <code class="computeroutput"><span class="identifier">Integer</span></code>
        result.
      </p>
<p>
        Returns <code class="computeroutput"><span class="identifier">result</span> <span class="special">=</span>
        <span class="identifier">a</span> <span class="special">*</span> <span class="identifier">b</span></code> without overflow or loss of precision
        in the multiplication.
      </p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Integer</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">I2</span><span class="special">&gt;</span>
<span class="identifier">Integer</span><span class="special">&amp;</span> <span class="identifier">add</span><span class="special">(</span><span class="identifier">Integer</span><span class="special">&amp;</span> <span class="identifier">result</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">I2</span><span class="special">&amp;</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">I2</span><span class="special">&amp;</span> <span class="identifier">b</span><span class="special">);</span>
</pre>
<p>
        Adds two <code class="computeroutput"><span class="identifier">I2</span></code> values, to produce
        a wider <code class="computeroutput"><span class="identifier">Integer</span></code> result.
      </p>
<p>
        Returns <code class="computeroutput"><span class="identifier">result</span> <span class="special">=</span>
        <span class="identifier">a</span> <span class="special">+</span> <span class="identifier">b</span></code> without overflow or loss of precision
        in the addition.
      </p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Integer</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">I2</span><span class="special">&gt;</span>
<span class="identifier">Integer</span><span class="special">&amp;</span> <span class="identifier">subtract</span><span class="special">(</span><span class="identifier">Integer</span><span class="special">&amp;</span> <span class="identifier">result</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">I2</span><span class="special">&amp;</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">I2</span><span class="special">&amp;</span> <span class="identifier">b</span><span class="special">);</span>
</pre>
<p>
        Subtracts two <code class="computeroutput"><span class="identifier">I2</span></code> values,
        to produce a wider <code class="computeroutput"><span class="identifier">Integer</span></code>
        result.
      </p>
<p>
        Returns <code class="computeroutput"><span class="identifier">result</span> <span class="special">=</span>
        <span class="identifier">a</span> <span class="special">-</span> <span class="identifier">b</span></code> without overflow or loss of precision
        in the subtraction.
      </p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Integer</span><span class="special">&gt;</span>
<span class="identifier">Integer</span> <span class="identifier">powm</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Integer</span><span class="special">&amp;</span> <span class="identifier">b</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Integer</span><span class="special">&amp;</span> <span class="identifier">p</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Integer</span><span class="special">&amp;</span> <span class="identifier">m</span><span class="special">);</span>
</pre>
<p>
        Returns b<sup>p</sup> % m.
      </p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Integer</span><span class="special">&gt;</span>
<span class="keyword">void</span> <span class="identifier">divide_qr</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Integer</span><span class="special">&amp;</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Integer</span><span class="special">&amp;</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">Integer</span><span class="special">&amp;</span> <span class="identifier">q</span><span class="special">,</span> <span class="identifier">Integer</span><span class="special">&amp;</span> <span class="identifier">r</span><span class="special">);</span>
</pre>
<p>
        Sets <code class="computeroutput"><span class="identifier">q</span> <span class="special">=</span>
        <span class="identifier">x</span> <span class="special">/</span> <span class="identifier">y</span></code> and <code class="computeroutput"><span class="identifier">r</span>
        <span class="special">=</span> <span class="identifier">x</span> <span class="special">%</span> <span class="identifier">y</span></code>.
      </p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Integer1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Integer2</span><span class="special">&gt;</span>
<span class="identifier">Integer2</span> <span class="identifier">integer_modulus</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Integer1</span><span class="special">&amp;</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">Integer2</span> <span class="identifier">val</span><span class="special">);</span>
</pre>
<p>
        Returns x % val;
      </p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Integer</span><span class="special">&gt;</span>
<span class="keyword">unsigned</span> <span class="identifier">lsb</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Integer</span><span class="special">&amp;</span> <span class="identifier">x</span><span class="special">);</span>
</pre>
<p>
        Returns the (zero-based) index of the least significant bit of <code class="computeroutput"><span class="identifier">x</span></code>.
      </p>
<p>
        Throws a <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">domain_error</span></code> if <code class="computeroutput"><span class="identifier">x</span>
        <span class="special">&lt;=</span> <span class="number">0</span></code>.
      </p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Integer</span><span class="special">&gt;</span>
<span class="keyword">unsigned</span> <span class="identifier">msb</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Integer</span><span class="special">&amp;</span> <span class="identifier">x</span><span class="special">);</span>
</pre>
<p>
        Returns the (zero-based) index of the most significant bit of <code class="computeroutput"><span class="identifier">x</span></code>.
      </p>
<p>
        Throws a <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">domain_error</span></code> if <code class="computeroutput"><span class="identifier">x</span>
        <span class="special">&lt;=</span> <span class="number">0</span></code>.
      </p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Integer</span><span class="special">&gt;</span>
<span class="keyword">bool</span> <span class="identifier">bit_test</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Integer</span><span class="special">&amp;</span> <span class="identifier">val</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">index</span><span class="special">);</span>
</pre>
<p>
        Returns <code class="computeroutput"><span class="keyword">true</span></code> if bit <code class="computeroutput"><span class="identifier">index</span></code> is set in <code class="computeroutput"><span class="identifier">val</span></code>.
      </p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Integer</span><span class="special">&gt;</span>
<span class="identifier">Integer</span><span class="special">&amp;</span> <span class="identifier">bit_set</span><span class="special">(</span><span class="identifier">Integer</span><span class="special">&amp;</span> <span class="identifier">val</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">index</span><span class="special">);</span>
</pre>
<p>
        Sets the <code class="computeroutput"><span class="identifier">index</span></code> bit in <code class="computeroutput"><span class="identifier">val</span></code>.
      </p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Integer</span><span class="special">&gt;</span>
<span class="identifier">Integer</span><span class="special">&amp;</span> <span class="identifier">bit_unset</span><span class="special">(</span><span class="identifier">Integer</span><span class="special">&amp;</span> <span class="identifier">val</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">index</span><span class="special">);</span>
</pre>
<p>
        Unsets the <code class="computeroutput"><span class="identifier">index</span></code> bit in
        <code class="computeroutput"><span class="identifier">val</span></code>.
      </p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Integer</span><span class="special">&gt;</span>
<span class="identifier">Integer</span><span class="special">&amp;</span> <span class="identifier">bit_flip</span><span class="special">(</span><span class="identifier">Integer</span><span class="special">&amp;</span> <span class="identifier">val</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">index</span><span class="special">);</span>
</pre>
<p>
        Flips the <code class="computeroutput"><span class="identifier">index</span></code> bit in <code class="computeroutput"><span class="identifier">val</span></code>.
      </p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Integer</span><span class="special">&gt;</span>
<span class="identifier">Integer</span> <span class="identifier">sqrt</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Integer</span><span class="special">&amp;</span> <span class="identifier">x</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Integer</span><span class="special">&gt;</span>
<span class="identifier">Integer</span> <span class="identifier">sqrt</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Integer</span><span class="special">&amp;</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">Integer</span><span class="special">&amp;</span> <span class="identifier">r</span><span class="special">);</span>
</pre>
<p>
        Returns the integer square root <code class="computeroutput"><span class="identifier">s</span></code>
        of x and sets <code class="computeroutput"><span class="identifier">r</span></code> to the remainder
        <span class="emphasis"><em>x - s<sup>2</sup></em></span>.
      </p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Engine</span><span class="special">&gt;</span>
<span class="keyword">bool</span> <span class="identifier">miller_rabin_test</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">number</span><span class="special">-</span><span class="keyword">or</span><span class="special">-</span><span class="identifier">expression</span><span class="special">-</span><span class="keyword">template</span><span class="special">-</span><span class="identifier">type</span><span class="special">&amp;</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">trials</span><span class="special">,</span> <span class="identifier">Engine</span><span class="special">&amp;</span> <span class="identifier">gen</span><span class="special">);</span>
<span class="keyword">bool</span> <span class="identifier">miller_rabin_test</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">number</span><span class="special">-</span><span class="keyword">or</span><span class="special">-</span><span class="identifier">expression</span><span class="special">-</span><span class="keyword">template</span><span class="special">-</span><span class="identifier">type</span><span class="special">&amp;</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">trials</span><span class="special">);</span>
</pre>
<p>
        The regular Miller-Rabin functions in <code class="computeroutput"><span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">multiprecision</span><span class="special">/</span><span class="identifier">miller_rabin</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></code>
        are defined in terms of the above generic operations, and so function equally
        well for <a href="https://en.cppreference.com/w/cpp/language/types" target="_top">fundamental
        (built-in) types</a> and multiprecision types.
      </p>
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      Maddock and Christopher Kormanyos<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
        file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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